Intereting Posts

Prime factor of $A=14^7+14^2+1$
limit of reciprocal of function
Relation between AB and BA
On a non-standard approach to the classification of conics?
Complex part of a contour integration not using contour integration
Existence of universal enveloping inverse semigroup (similar to “Grothendieck group”)
What does the $\phi^{-1}$ mean in this formula: $\rho_y(y)=\rho_x(\phi^{-1}(y))\left|\frac{{d}\phi^{-1}}{{d}y}\right|$?
Can a group have more subgroups than it has elements?
How to prove $\sum_{n=0}^{\infty} \frac{n^2}{2^n} = 6$?
Unique perpendicular line
How to prove that $\lim_{n \to \infty} n x^{n} = 0 $ when $0<x<1$?
What are some examples of coolrings that cannot be expressed in the form $R$?
Is there anything like “cubic formula”?
Norm of a Kernel Operator
Translation invariant measures on $\mathbb R$.

Suppose I embed a manifold-with-boundary $M$ in some $\mathbb{R}^n$. Are there conditions (necessary, sufficient, or both) that can help determine when the topological boundary of $M$ is equal to the manifold boundary?

By “topological boundary,” I’m referring to $\text{Bd } M$, which is the closure minus the interior (relative to $\mathbb{R}^n$).

By “manifold boundary,” I mean the boundary $\partial M$ that is specified in the definition of “manifold-with-boundary.”

- Vector Field in a complex projective space
- The injectivity of torus in the category of abelian Lie groups
- How to show that the geodesics of a metric are the solutions to a second-order differential equation?
- $k$-jet transitivity of diffeomorphism group
- Looking For a Neat Proof of the Fact that the Grassmannian Manifold is Hausdorff
- Exponential map on the the n-sphere

- De Rham Cohomology of $M \times \mathbb{S}^1$
- Complex structure on $\mathbb{C}\mathbb{P}^2\# \dots \# \mathbb{C}\mathbb{P}^2$
- Holomorphic Poincaré conjecture
- A covering map from a differentiable manifold
- Are there Kirby diagrams for manifolds with boundaries?
- Prove that curve with zero torsion is planar
- Relationship beween Ricci curvature and sectional curvature
- $SL(n)$ is a differentiable manifold
- Hopf's theorem on CMC surfaces
- Visualizing Frobenius Theorem

If you embed a smooth $m$-manifold $M$ smoothly in $\mathbb{R}^n$

then the “topological boundary” of $M$ is the closure of $M$. As

locally each point of $M$ has a neighbourhood in $\mathbb{R}^n$

where $M$ looks like $\mathbb{R}^m$, then no point of $M$ is interior.

When $m=n$ and $M$ is compact then, yes (at least in the smooth case)

the topological and manifold boundaries coincide.

I expect the above hold for topological embeddings but won’t swear

to it; they need not be locally flat (see nasties like the Alexander

horned sphere),

- solve $\ln(n!) = \Theta(n\ln(n))$ without stirling approximation
- Probabilistic Proof of $\prod\limits_{i=1}^\infty\cos\left(\frac t{2^i}\right)=\frac{\sin t}t$
- Infinite algebraic extension of a finite field
- Modules over $k$
- Matrices – Conditions for $AB+BA=0$
- If $a$ is even and $b$ is odd then $\gcd(2^{a}+1,2^{b}+1)=1$
- Division algorithm of multivariate polynomial
- Why is the volume of a parallelepiped equal to the square root of $\sqrt{det(AA^T)}$
- Question on the presentation of $(\mathbb{R}, +)$
- Conjectured new primality test for Mersenne numbers
- How many times more than $0$?
- Analogy of ideals with Normal subgroups in groups.
- Solving the SDE $dX(t) = (c(t) + d(t)X(t))dt + (e(t) + f(t)X(t))dW(t)$
- Why use geometric algebra and not differential forms?
- More Theoretical and Less Computational Linear Algebra Textbook